How can I export UV Baked Image Filmic Log Encoding Layer (Save as Image)
By changing color space you don't actually change the image itself but rather the view of it, same image seen differently.
If you want to bake the view in the texture you need to check save as render when you save your image.
  

But what is it for? Will your texture will be lighted in the other program?
If it's the case it's isn't a good aproach, better bake it as an Albedo.
The transfer fonction of the filmic from scene referred data to display referred data must be the last operation on the image. It means that if your texture will be light after Blender you would need filmic in the other software.  

EDIT1  

You're not in the right place to manage color, here it's just the color space of the texture when you look at it.
To save your image with the right look you must :  

-Go in the scene tab
-Under color management set display device to sRGB, view to Filmic Log, and look choose the one you want.
-After that you can save your image.  

  

EDIT2  

I think I have the point now, you are confused by what happens in the image editor. When you rendered your texture and saved it, you already baked the filmic look in the texture, then you load the texture in the image editor and change the color space to filmic.
What is happening in image editor when you change the color spaces to Filmic is that it bring back the transformation to linear (only if you baked your image with filmic).  

The steps you made:


Linear scene referred datas (raw render) to BT.709/Filmic Log color space (saved image)
BT.709/Filmic Log back to linear scene referred data.
Consider the image color space in the image editor a invertor of thansformation curves.


Like this you can encode a linear scene referred data (that you can normally only export in EXR 32bit) in a 8/16bit image (with the possibility to decode back to linear scene referred data) exept for the desaturation part of the Filmic. I think to decode back without loss you'll need 16bit image.  

So for your need, bake your image with the Filmic (like I said in the Edit1 part), then put it in your other software (emission shader or so, strength to 1) you're done.  

And if you want to see a texture in image editor let it by default unless you know why you change it. In your case you must set it to sRGB EOTF.